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 A319564 Number of T_0 integer partitions of n. 30
 1, 1, 2, 3, 5, 7, 10, 14, 21, 29, 40, 53, 73, 95, 128, 168, 221, 282, 368, 466, 599, 759, 962, 1201, 1513, 1881, 2345, 2901, 3590, 4407, 5416, 6614, 8083, 9827, 11937, 14442, 17458, 21021, 25299, 30347, 36363, 43438, 51843, 61705, 73384, 87054, 103149, 121949 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. For an integer partition the T_0 condition means the dual of the multiset partition obtained by factoring each part into prime numbers is strict (no repeated blocks). Also the number of integer partitions of n with no equivalent primes. In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent. See A316978 for more examples. LINKS MATHEMATICA primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]] dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}] Table[Length[Select[IntegerPartitions[n], UnsameQ@@dual[primeMS/@#]&]], {n, 20}] CROSSREFS Cf. A000009, A000041, A001970, A007716, A059201, A305148, A316978, A316979, A316983, A319558, A319616, A319728. Sequence in context: A002062 A005688 A241550 * A221943 A120446 A082531 Adjacent sequences:  A319561 A319562 A319563 * A319565 A319566 A319567 KEYWORD nonn AUTHOR Gus Wiseman, Sep 23 2018 STATUS approved

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)